Integral representations and integral transforms of some families of ...

12 N. Elezović et al.
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552
553
554
555
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558
559
560
561
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and
F c
(
S
(α,β)
μ (r;{k 2/α }) ) (x) =
∫ ∞
0
cos(xr)S μ
(α,β) (r;{k 2/α })dr
√ π · 2
3/2−μ
=
Ɣ(μ)
(
x>0; μ>max
∫ ∞
t μ−1/2
x
where the -functions are given by (53), (54) and (55) with, of course,
a = x 2 ,b= t 2 ,α= μ, β = μ − β α
(
μ, μ − β α ,μ− β α + 1 2 ; x 2 , t )
dt
2
e t − 1 c
{ 2β
α , β α + 1 })
, (57)
2
and γ = μ − β α + 1 2 . (58)
In their special case when β = 0, our last results (55) and (56) would reduce immediately to
(49) and (50), respectively.
4. The Mellin transforms
I. In order to evaluate the Mellin transforms of S(r) and ˜S(r), we apply the familiar integral
formula [6, Vol. I, p. 68, Entry 2.3 (1)]:
We thus obtain
and
∫ ∞
0
sin t
t ν
M(S(r))(s) =
dt = Ɣ(1 − ν)cos
∫ ∞
0
∫ ∞
r s−1 S(r)dr =
( πν
) ( )
0 < R(ν) < 2 . (59)
2
∫ ∞
(∫
t ∞
sin(rt)
e t − 1 0 r 2−s
0
( 1
r s−1 r
)
dr dt
∫ ∞
=
0
( π
) ∫ ∞
= Ɣ(s − 1) cos
2 (2 − s) t
0 e t − 1 dt
( πs
)
=−Ɣ(s − 1)Ɣ(2)ζ(2) cos
2
=
π 3 ( πs
)
12Ɣ(2 − s) csc 2
M( ˜S(r))(s) =−Ɣ(s − 1) cos
0
t sin(rt)
e t − 1 dt )
dr
(0 < R(s) < 2) (60)
( πs
) ∫ ∞
2
=−Ɣ(2)η(2)Ɣ(s − 1) cos
=
π 3
24Ɣ(2 − s)
( πs
)
2
0
t
e t + 1 dt
)
( πs
2
(0 < R(s) < 2), (61)